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Let $\{ \cdot \}$ denote the fractional part function. Given $a, b, c, d \in \mathbb{R}$ such that $a \leqslant b$, $c \leqslant d$, $b - a = d - c$ and $\{ a x \} - \{ b x\} = \{c x \} - \{d x \}$ for $x \in \mathbb{Z}_{\geqslant 0}$, does it necessarily follow that there is a $\delta \in \mathbb{Z}$ such that $c = a + \delta$ and $d = b + \delta$? More generally, if $\{ a x \} - \{ b x\} = \{c x \} - \{d x \}$ for $x \in \mathbb{R}_{\geqslant 0}$, does it necessarily follow that $c = a$ and $d = b$?

I believe the answer is 'yes' in both cases, but I'm not quite sure how to prove it. If the constants are rational, then I believe a proof could follow by considering periodicity. Otherwise, if the constants are irrational, maybe a density argument would suffice.

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Do you mean for some $x$ or for all $x$? –  N. S. Apr 24 '12 at 18:24
    
Neither. The first case is for $x \in \mathbb{Z}_{\geqslant 0}$, the second for $x \in \mathbb{R}_{\geqslant 0}$. –  user02138 Apr 24 '12 at 18:36

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up vote 4 down vote accepted
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Neither of these is true as stated. The second one can easily be fixed, and I suspect the first one might be salvaged, too.

For the first one, the case where $b-a=d-c$ is an integer is a counterexample, since in this case $\{ax\}-\{bx\}=\{cx\}-\{dx\}=0$ for all $x\in\mathbb Z_{\ge0}$ regardless of whether $c-a=d-b$ is an integer. I suspect it might be true if you exclude this case, though.

For the second one, $a=b$ and $c=d$ is still a counterexample, but you can easily exclude that by requiring $a\lt b$ and $c\lt d$ instead. Then it does indeed follow that $c=a$ and $d=b$.

If a unique one of the numbers, say $d$, has highest absolute value, the jump at $x=1/|d|$ isn't cancelled, so the equation can't hold. That leaves the following cases to consider:

  • $d=-a$: In this case $b=-c$, and the two sums $\{ax\}+\{-ax\}$ and $\{bx\}+\{-bx\}$ would cause spikes at different positions that don't cancel.
  • $d=-b$: In this case either $a$ or $c$ would have greater absolute value.
  • $d=-c$: In this case either $a$ or $b$ would have greater absolute value.
  • $d=a$: In this case either $b$ or $c$ would have greater absolute value.

That leaves only the possibility $d=b$, and thus $c=a$.

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